MPhys Physics with Astrophysics

Random Processes in Physics

To introduce and develop the mathematical skills and knowledge needed to understand and use probability theory in physics.

On completion of the course, students will be able to: 

1. Be cognizant of and be able use appropriately, the fundamentals of probability theory.
2. Set up and solve models of physical processes involving randomness.
3. Be aware of and be able to critically apply, some of the important probability distributions that are used by physicists.

1.  Elements of probability                                                                                        

• Introduction:  What is probability?
• How to calculate probabilities:  permutations and combinations
• Conditional probability

 2.  Probability distributions                                                                                     

• Discrete random variables; expectation value and variance
• Example:  the geometric distribution
• Continuous random variables; the probability density function
• Examples:  the uniform distribution; the normal (or Gaussian) distribution

 3. Exponential Probability Distribution                                                                    

• Probability of radioactive decay
• Probability of collisions in a gas; mean free path
• Generalisation: “hazard rate” and survival probability

 4. Poisson Probability Distribution                                                                           

• Probability of occurrence of n random events
• Properties of the Poisson distribution
• Gaussian limit of the Poisson distribution

 5. Binomial Probability Distribution                                                            

• Binomial distribution for n trials
• Irreversible expansion of a gas
• Poisson and Gaussian limits of the binomial distribution
• Random walks and diffusion

Feedback will be provided through self assessed problems, and on-line assessment of weekly examples sheets. General feedback will also be given during the weekly Q&A session.

A suitable introduction to probability can be found in:

Chapters 39 and 40 of Mathematical Techniques, 3rd edition, Jordan, D. & Smith, P.

Chapters 20 & 21 of Mathematics for Engineers and Scientists, Weltner, K., Gorsjean, J., Schuster, P. & Weber, W.

Chapter 3 of Statistics, Barlow, R.J.